An exact version of Shor's order finding algorithm

We present an efficient exact version of Shor's order finding algorithm when a multiple m of the order r is known. The algorithm consists of two main ingredients. The first ingredient is the exact quantum Fourier transform proposed by Mosca and Zalka. The second ingredient is an amplitude amplification version of Brassard and Hoyer combined with some ideas from the exact discrete logarithm procedure by Mosca and Zalka.